Journal of Computational Physics 225 (2007) 449–463
www.elsevier.com/locate/jcp
Spectral method for matching exterior and interior
elliptic problems
Piotr Boronski
Center for Turbulence Research, Stanford University, Bldg. 500, Stanford, CA 94305-3035, United States
Received 22 August 2006; received in revised form 7 December 2006; accepted 11 December 2006
Available online 22 December 2006
Abstract
A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method
is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their
normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the
interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a
preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green’s function with a surface density; this
surface density is itself the solution to an integral equation which has an analytic solution when the boundary values
are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic
functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic
potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is
discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum.
2006 Elsevier Inc. All rights reserved.
Keywords: Influence matrix; Spectral method; Chebyshev polynomials; Boundary integral method; Magnetohydrodynamics; Green’s
functions; Harmonic functions; Laplace’s equation; Exterior problem
1. Motivation
The search for a self-sustaining magnetohydrodynamic dynamo has taken on great momentum in recent
years, as researchers have sought to produce dynamos in the laboratory [1–5] and in simulations [6–16].
One of the fundamental problems in numerical magnetohydrodynamics is the formulation of boundary conditions. The governing equations describe the velocity and magnetic field in a finite container of electrically
conducting fluid. At the container boundaries, the velocity is specified, but the magnetic field is not. Instead,
E-mail address: boronski@gmail.com
0021-9991/$ - see front matter 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcp.2006.12.005
450
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
the magnetic field is required to satisfy continuity conditions with the exterior magnetic field in the domain
surrounding the fluid. The nature of these conditions depends on the properties of the surrounding medium;
a complete discussion can be found in [17].
Under the quasi-static approximation [17], for a given velocity field u and magnetic Reynolds number Rm,
the equations describing the interior magnetic field are
ot B ¼ r ðu BÞ þ
1
DB
Rm
rB¼0
ð1aÞ
ð1bÞ
The case of a fluid of finite electric conductivity restricted to a finite volume and surrounded by vacuum is of
special importance because it models a number of experimental, geophysical, and astrophysical configurations. Since there are no electrical currents in a vacuum, Bvac is curl-free, and is therefore the gradient of a
potential if the exterior domain is simply connected. The exterior magnetic field then obeys:
Bvac ¼ r/vac
ð2aÞ
vac
D/ ¼ 0
Bvac ! 0
jxj ! 1
ð2bÞ
ð2cÞ
but is otherwise not fixed. The magnetic field is required to be continuous at the boundary:
B Bvac ¼ 0
x 2 oX
ð3Þ
In this case, continuity of all three components of the magnetic field are sufficient to uniquely determine both
the interior and exterior fields. Our ultimate goal is to transform (2)–(3) into boundary conditions that can be
applied to (1) without calculating Bvac. The general principle we will employ is to construct a complete basis of
exterior solutions Bvac of (2) in a preprocessing step, and to calculate BjoX for each member of the basis. The
matching conditions (3) will then yield boundary conditions for B.
To explore this approach, we will apply it to the simpler analogous scalar problem of the Poisson problem
in an interior domain. We will require the solution to match continuously to an exterior solution satisfying
Laplace’s equation. In this case, both Dirichlet and Neumann matching conditions are necessary to specify
a unique solution. Formally, we wish to solve the following problem:
DU ¼ q in X
ð4aÞ
D/ ¼ 0
ð4bÞ
outside X
with boundary conditions:
UðxÞ /ðxÞ ¼ 0; x 2 oX
on UðxÞ on /ðxÞ ¼ 0; x 2 oX
ð5aÞ
ð5bÞ
r/ðxÞ ! 0;
ð5cÞ
jxj ! 1
where X is a bounded domain with boundary oX. A physical interpretation of (4)–(5) is that of an electrostatic
potential U of a field generated by charges distributed in space with the density q, where the electrical permeability of the vacuum is taken to be one by the choice of units. We wish to calculate the interior solution U
without explicitly constructing the exterior solution /.
There exists a vast literature on the numerical solution of the fundamental physical problems (4)–(5) and
(1)–(3). We will briefly survey a small portion of this literature here, and postpone a more detailed comparison
between our method and others to a later section.
The main tool by which exterior domains can be eliminated is Green’s theorem, which replaces elliptic
differential equations over a domain with integrals over the bounding surfaces. The use of methods based
on boundary integrals has grown explosively since the 1970–1980s to solve engineering problems from fields
such as acoustics, elasticity, electromagnetism and fluid mechanics [18–26]. Hybrid methods, coupling a differential equation formulation in a domain and a boundary integral formulation at the boundary via an
influence matrix, were also developed at the same time to solve (4)–(5) and similar problems. The majority
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
451
of these approaches have been based on finite elements and are hence applicable to complicated real-world
geometries. The boundary integrals are discretized with techniques derived from finite element theory, leading to the term boundary element method, and the hybrid methods use finite elements to solve the equations
in the domain.
This situation contrasts with magnetohydrodynamics, which has been dominated by spectral methods.
Spherical domains are standard, for geophysical and astrophysical reasons. Spectral methods can then be
based on spherical harmonics and the poloidal–toroidal decomposition [7–10]. The solution to Laplace’s equation on the exterior is immediate and, moreover, solutions and associated boundary conditions for each spherical harmonic and toroidal or poloidal component are decoupled. Boundary conditions at the interface can
then be formulated for each mode without the use of an influence matrix.
The technique which we will describe is based on spectral methods, but the geometry is assumed to be somewhat more complicated. Our technique occupies the niche which spectral methods staked out in the 1970–
1980s when the use of Chebyshev polynomials became commonplace [27,28] to represent domains with one
or more non-periodic directions. In keeping with this tradition, we expect its main application to be to tensor-product domains whose boundaries consist of a small number of piecewise-smooth surfaces, such as
the finite three-dimensional cylinder which is our eventual goal.
We also mention here some other recent approaches to solving the magnetohydrodynamic equations [12–
16], with a view to generalizing the geometry and/or increasing parallelization. In [13,14], a finite volume
method is used to discretize the solution in the interior, which is matched to that in the exterior vacuum
via a boundary element method. [15] describes an integral equation formulation for the entire domain, and
[16] uses finite elements with a penalty method to apply boundary conditions.
2. Influence matrix formulation
We formulate a two-stage method for solving (4)–(5), consisting of an initial preprocessing step which
depends only on the geometry, followed by a step whereby solutions for many different distributions q
can be generated at little incremental cost. This is the usual description of the decomposition of U into
homogeneous and particular solutions, with the additional proviso that solutions in the exterior domain
are to be taken into account in the preparation of the homogeneous solutions. We will construct the homogeneous solutions by generating harmonic bases fUhj g and f/j g of interior and exterior solutions, corresponding to Dirichlet boundary data ffj g to be specified later. We decompose (4)–(5) into the Poisson
and Laplace problems:
DUp ¼ q
DUhj ¼ 0
in X;
p
Uhj joX
U joX ¼ 0;
in X;
¼ fj
D/j ¼ 0
outside X;
/j joX ¼ fj ;
ð6Þ
r/j j1 ¼ 0
and then construct the linear superpositions:
X
X
U ¼ Up þ Uh ;
Uh ¼
cj Uhj ;
/¼
cj / j
j
Then
DU ¼ DUp þ
D/ ¼
X
X
cj DUhj ¼ q
in X
ð8Þ
j
cj D/j ¼ 0
j
ðU /ÞjoX ¼ Up joX þ
r/j1 ¼
ð7Þ
j
X
outside X
X
cj ðUhj /j ÞjoX ¼ 0
ð9Þ
ð10Þ
j
cj r/j j1 ¼ 0
j
are already satisfied by construction, while
ð11Þ
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P. Boronski / Journal of Computational Physics 225 (2007) 449–463
on ðU /ÞjoX ¼ on Up joX þ
X
j
cj on ðUhj /j Þ
oX
¼0
ð12Þ
constitutes a system of equations to be solved for cj, where the derivative with respect to the normal is taken in
the direction from the interior to the exterior region for both U and /. U is then set equal to the sum in (7). If
the interior harmonic functions are not stored, U can be obtained by solving:
DU ¼ q in X
X
UjoX ¼
cj f j
ð13Þ
ð14Þ
j
Using xi to index points on the boundary, (12) can be discretized as:
X
½on ð/j Uhj Þðxi Þcj ¼ on Up ðxi Þ
ð15Þ
j
Eq. (15) shows that the goal of the preprocessing step is the construction and inversion of the influence or
capacitance matrix:
C ij ½on ð/j Uhj Þðxi Þ
ð16Þ
The functions ffj g are required to constitute a complete set for values along the discretized boundary oX. Another way to describe the influence matrix is as a discrete representation of the difference between the Dirichlet-to-Neumann mappings in the exterior and in the interior regions.
Equivalently, on ð/j Uhj ÞjoX can be represented as coefficients of a basis set fgi g (which may be identical
with the set of boundary value functions ffi g) along each boundary. Eq. (12) is then discretized as:
X
hon ð/j Uhj Þ; gi icj ¼ hon Up ; gi i
ð17Þ
j
Although we will use xi and the notation in (15) in what follows, the method is easily reformulated using
(17).
3. Solution of Poisson and Laplace problems
3.1. Interior domain
We now turn to the solution of (6). For the interior problems listed in the first two columns, we assume that
we dispose of a solver able to compute solutions to Poisson’s equation in X with any specified boundary values. In principle, any numerical method can be used. In our particular case, we use a spectral discretization
[28]
Uðx; yÞ ¼
K;L
X
Tk ðx=H ÞTl ðyÞ
ð18Þ
k;l¼0
for the rectangle ½H ; H ½1; 1. The spectral basis functions are the Chebyshev polynomials
Tk ðxÞ ¼ cosðk arccosðxÞÞ. Taking H P 1, we set K P L. We use a standard method [28] to solve the Poisson
equation with Dirichlet boundary conditions, diagonalizing the discretized second derivative operator in y,
and using recursion relations to treat the second derivative in x.
3.2. Exterior harmonic functions
Our main focus is on the construction of the exterior harmonic solutions, specified in the third column of
(6). In order to avoid truncating or spatially discretizing the exterior domain, we will construct f/j g using the
fundamental solution of the Laplace equation: the Green’s function satisfying
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
D0x Gðx; x0 Þ ¼ dðx x0 Þ
rGðx; x0 Þ ¼ 0
for x0 ! 1
453
ð19Þ
ð20Þ
For a specified boundary value distribution f(x), we first calculate an appropriate source distribution r(x) on
the boundary by solving the integral equation:
Z
Gðx; x0 Þrðx0 Þ ¼ f ðxÞ for x 2 oX
ð21Þ
oX
The exterior harmonic function /(x) required is then:
Z
/ðxÞ Gðx; x0 Þrðx0 Þ
ð22Þ
oX
where x takes values either on or off oX.
We now apply (21) and (22) to our particular test problem of a rectangle. We divide the set of boundary
distributions into four sets, each taking non-zero values on only one side of the rectangle. In two dimensions,
the fundamental Green’s function solving (20) is
1
ln jx x0 j
2p
Eq. (21) thus reduces to:
Z b
1
ln jx x0 jrðx0 Þ dx0 ¼ f ðxÞ
2p
a
Eq. (24) is known as Symm’s or Carleman’s equation and has the following solution [29,30]:
#
"Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z b
b
ðx0 aÞðb x0 Þf 0 ðx0 Þ dx0
2
1
f ðx0 Þ dx0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ
lnððb aÞ=4Þ a
x0 x
p ðx aÞðb xÞ a
ðx0 aÞðb x0 Þ
ð23Þ
ð24Þ
ð25Þ
Rifb b a 6¼ 4. (If b a1 ¼ 4 then the second integral in (25) can be replaced by an arbitrary constant if
f ðtÞ½ðt aÞðb tÞ 2 dt ¼ 0.)
a
Up to now, we have not specified the Dirichlet boundary values f. The choice of boundary value distributions is restricted only by the requirement, stated in Section 2, that the set of distributions form a basis for
functions defined on the boundary oX. Because we use Chebyshev polynomials to represent the interior solutions, it is convenient to take as boundary values fk(x) each of the functions Tk ðx=H Þ on the interval ½H ; H .
The corresponding solutions rk(x;H) obtained from evaluating (25) are:
2pk;
k>0
Tk ðx=H Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rk ðx; H Þ ¼ Ak
¼
;
A
k
1
ð26Þ
x 2
2p½lnðH
=2Þ
;
k
¼0
pH 1 H
This remarkable property – the fact that weighted Chebyshev polynomials are also obtained as the source
distributions rk(x) when the boundary values fk(x) are Chebyshev polynomials – is related to the very reason pthat
Chebyshev
polynomials are optimal in approximating polynomials on the interval. The function
ffiffiffiffiffiffiffiffiffiffiffiffi
ffi
1=p 1 x2 in (26) (for H ¼ 1) is the weight with respect to which Chebyshev polynomials are orthogonal
on the interval and is the asymptotic density of the Chebyshev interpolation points cosðpj=J Þ, the extrema
of the Chebyshev polynomials. See [31,32] for further details. Note also that the orthogonality of the
Chebyshev polynomials with respect to this weight causes the second integral in (25) to vanish except
for T0 .
The corresponding harmonic functions /xk are constructed via
Z H
1
ln jx x0^ex jrk ðx0 ; H Þ dx0
ð27Þ
/xk ðxÞ ¼
2p
H
and are illustrated in Fig. 1. Specifying values along the segment ½1; 1 in the y direction, we obtain:
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P. Boronski / Journal of Computational Physics 225 (2007) 449–463
Fig. 1. Potentials /xk ðxÞ generated by line source distributions rk ðx; H ¼ 2Þ. Values of the potentials on the line segment x 2 ½1; 1
correspond to the Chebyshev polynomials Tk ðxÞ. From top to bottom: r0(x), r1(x), r4(x).
/yl ðxÞ
¼
Z
1
1
1
ln jx y 0^ey jrl ðy 0 ; 1Þ dy 0
2p
ð28Þ
Note that rk ðx; H Þ ¼ rk ðx=H ; 1Þ=H for k > 0 and r0 ðx; H Þ ¼ Hlnð1=2Þ
r ðx=H ; 1Þ.
lnðH =2Þ 0
The harmonic functions corresponding to specified values along the lower or upper boundaries (y ¼ 1,
jxj < H ) are /xk ðx 1^ey Þ; those corresponding to the left or right boundaries (x ¼ H , jyj < 1) are
/yl ðx H^ex Þ. We do not require the functions /xk , /yl either inside or outside X, but only the values and normal
derivatives on the boundary. Although, for example, the values of /xk ðx þ 1^ey Þ on the lower boundary are
merely the specified values fk(x), its values on the other three boundaries must be calculated via (27). When
evaluating the normal derivatives, the kernel Gðx; x0 Þ is differentiated before integration:
Z H
Z H
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
y
2
x
0
0
0
2
ln ðx x Þ þ y rk ðx ; H Þ dx ¼
rk ðx0 ; H Þ dx0
ð29Þ
oy /k ðx; yÞ ¼ oy
0
2p
2p
ðx x Þ2 þ y 2
H
H
and similarly for ox /yl .
Any exterior harmonic function can be approximated by the truncated series:
/K;L ðxÞ ¼
K 1
X
k¼0
x
x
½cx;
ex Þ þ cx;þ
ex Þ þ
k /k ðx þ H^
k /k ðx H^
L1
X
l¼0
y
y
½cy;
ey Þ þ cy;þ
ey Þ
l /l ðx þ 1^
l /l ðx 1^
ð30Þ
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
455
x;þ y; y;þ
The potential /(x) of (30) is defined by the 2ðK þ LÞ coefficients fcx;
k ; ck ; cl ; cl g. A very important propy
x
erty of the harmonic basis f/k ðxÞ; /l ðxÞg is that it represents a near-boundary field uniformly. This means that
the truncated series (30) converges uniformly for any smooth boundary data and for all locations x near the
boundary. This is a direct consequence of the excellent convergence properties of Chebyshev approximation
applied to (24); a proof can be found in [33]. This property does not necessarily hold for other harmonic bases,
in particular spherical harmonics, for which near-boundary convergence cannot be achieved, leading to a
strong Gibbs effect.
In the taxonomy of boundary integral methods, equations (21) and (22) constitute an indirect method, in
that the intermediate surface charge density r is constructed; this is done by solving the Fredholm integral
equation of the first kind (24). The surface charge density is a single-layer rather than a double-layer (dipole)
potential; equivalently G, rather than oG=on, is used in the representation. Because the method determines
only the boundary values and normal derivatives of the exterior solution, it is not the preferred approach when
the exterior potential is itself required at each time step: although the exterior solution can be sampled at any
location, this is computationally expensive, as is often the case for boundary integral methods.
4. Electrostatic example
We apply our method to a simple problem from electrostatics, the distribution of electric charges qm confined in a rectangular domain but localized around the origin:
(
2 2
rm er =d cosðmhÞ for jxj 6 1 and jyj 6 1
qm ðr; hÞ ¼
ð31Þ
0
for jxj > 1 or jyj > 1
~ m due to unbounded sources (not reThe rm factor in (31) ensures regularity of qm at r ¼ 0. The potential U
stricted to the interior domain) can be found analytically:
2 2
~ m¼0 ðr; hÞ ¼ d Ei 1; r þ 2 logðrÞ
U
ð32aÞ
4
d2
d4 r22
~
d
e 1 cos h
Um¼1 ðr; hÞ ¼
ð32bÞ
4r
4 r2
2
2
2 d 2
~ m¼2 ðr; hÞ ¼ d
e
U
d
þ
r
d
cos 2h
ð32cÞ
4r2
R1
~ m are chosen to be finite at r ¼ 0.
where Eiða; kÞ is the error function Eiða; kÞ 1 ekr ra dr and U
We seek the corresponding electric potential. The parameter d is chosen to make qm very small near the
boundaries. We expect the solution to be almost unaffected by the presence of boundaries. The source distribution qm¼0 ðr; hÞ should therefore lead to a potential which is almost axisymmetric. Fig. 2 shows the potential
obtained numerically for d2 ¼ 0:15 using the spectral resolution N ¼ 8 in both directions. The domain boundary is represented by a bold square. The contours are almost perfectly circular, as should be the case for d
small, showing that the presence of the boundaries has minimal effect.
To evaluate the error convergence of the method we computed the relative error Em(N) defined as
Em ðN Þ ¼ sup
r;h
~ m ðr; hÞ UN ðr; hÞj
jU
m
jUm ðr; hÞj
ð33Þ
where UNm ðr; hÞ is the solution computed numerically with spectral resolution N in both spatial directions of the
~ m ðr; hÞ is the analytic solution (32) in the absence of the bounding square. Fig. 3 proves
bounding square and U
the exponential convergence of the method.
N ¼16
¼16
¼16
Figs. 4–5 show the electric potentials Um¼1
and UNm¼2
for d ¼ 0:1. Fig. 6 shows UNm¼1
with d ¼ 2. In Fig. 7,
the dipole source distribution has been rotated by 45 about the origin. For this large value of d, charges are
located near the boundary. In each case with d 1, we observed exponential convergence toward solution
(32). Convergence can only be confirmed up to a limited precision since the analytic solution (32) does not
correspond exactly to the problem we are solving numerically, in which sources are confined to the interior
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P. Boronski / Journal of Computational Physics 225 (2007) 449–463
N ¼8
Fig. 2. Potential Um¼0
for d ¼ 0:15. Maximal relative error is Em¼0 ðN ¼ 8Þ 0:03.
Fig. 3. Convergence test: log10 Em¼0 ðN Þ plotted for N ¼ ½6; . . . ; 16, d2 ¼ 0:1.
square. The best agreement can be achieved for small values of d. If the numerical solution with highest spectral resolution (here N ¼ 64) is instead taken as a reference, then the method converges to this solution spectrally up to machine precision for any value of d.
5. Implementation
5.1. Summary and computation cost
We describe the implementation of the method for our illustrative example (18) of the rectangle
½H ; H ½1; 1 with double Chebyshev discretization ðK þ 1Þ ðL þ 1Þ.
The total preprocessing step consists of:
Evaluation of the values and the normal derivatives of the exterior harmonic solutions on the boundary.
Calculation of the interior harmonic solutions.
Inversion or LU decomposition of the influence matrix.
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
457
N¼16
Fig. 4. Potential Um¼1
for d ¼ 0:1.
N¼16
Fig. 5. Potential Um¼2
for d ¼ 0:1.
For each particular right-hand-side q, the operations consist of:
Solving for a single particular solution.
Acting with the inverse of the influence matrix.
Using the corrected Dirichlet boundary conditions to calculate the final solution.
The total number J of boundary points is 2ðK þ LÞ. The inversion or LU-decomposition of the influence
matrix C in the preprocessing stage requires a time proportional to J3, while each solution of the linear system
(15) determining the coefficients of the homogeneous solutions requires a time proportional to J2. Each interior solution is calculated at a cost proportional to KL2.
Symmetry can be used to reduce the cost of each step. The symmetry of the rectangle divides all the independent harmonic solutions into four mutually orthogonal classes. Decoupling the Laplacian operator
according to parity in x and y leads to four Poisson problems, each with resolution K/2 · L/2, thus reducing
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P. Boronski / Journal of Computational Physics 225 (2007) 449–463
N¼16
Fig. 6. Potential Um¼1
for d ¼ 2.
¼16
Fig. 7. UNm¼1
ðr; h þ p4Þ for d ¼ 2.
the time by a factor of two. Decoupling by parity also reduces the influence matrix C to four matrices, the
dimensions of which are one fourth of that of the original matrix.
Table 1 gives the operation count of each step, taking into account the reductions permitted by symmetry.
5.2. Singular integrals
The integrations in (27)–(29) are performed numerically. Special attention must be paid in doing so since
both the kernel Gðx; x0 Þ and the density rðx0 Þ have integrable singularities within the domain of integration.
The singular points are x0^ex ¼ x and x0 ¼ H for /xk and y 0^ey ¼ x and y 0 ¼ 1 for /yl . Dedicated adaptive
quadratures (see [34]) can be used to compute these integrals accurately.
It is also possible to evaluate the singular part of the integral analytically, reducing the numerical problem
to the evaluation of integrals with non-singular integrands. The remaining integrand is piecewise C 1 and can
be integrated with spectral precision over each of the regular subdomains. Singularity subtraction greatly
decreases the variation in grid density needed to sample the integrand homogeneously, thereby significantly
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
459
Table 1
Operation count of each preprocessing and right-hand-side-dependent step for a rectangle discretized with (K+1) · (L+1) Chebyshev
polynomials and points
Calculation
Result
Cost
Preprocessing
Exterior harmonic solutions
/j joX ; on /j joX
K ðIntx þ Int0x Þ þ L ðInty þ Int0y Þ
(K + L)KL2/4
Influence matrix inversion/decomposition
on Uhj
1
C
(K + L)3/2
For each right-hand-side
Particular solution
Action with influence matrix
Corrected solution
Up
cj
U
KL2/2
(K + L)2
KL2/2
Interior harmonic solutions
Intx and Int0x are the costs of performing the singular integrals over x in (27) and (29), and Inty and Int0y are those of the analogous integrals
over y.
accelerating the numerical integrations in (27)–(29). Specifically, an adaptive method requires a smaller number of iterations, or, alternatively, a non-adaptive method requires a coarser resolution. However, the convergence of our approximation with K, L is exponential (spectral), regardless of whether the singular part of the
integral is subtracted or included in the numerical evaluation.
5.3. Conditioning of matrices
The influence matrix (16) is not immediately invertible. Because of the redundancy of information at the
corners, this matrix has exactly four zero singular values or eigenvalues. The corresponding linear system
can be solved after arbitrarily correcting singular values or eigenvalues of the influence matrix; see [35,36]
for more details.
The condition number C of the corrected matrix depends on the spatial resolution N and the maximal order
of derivatives used to express the boundary conditions. In our case of a Neumann boundary condition and a
resolution N in each direction, the condition number scales as C ¼ OðN 2 Þ. Fitting the condition numbers computed for N 2 ½2; 32 with a parabola (see Fig. 8) yields a formula for predicting the condition number for an
arbitrary resolution:
Fig. 8. Quadratic fit of the condition number CðN Þ of the influence matrix defined in (16).
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P. Boronski / Journal of Computational Physics 225 (2007) 449–463
CðN Þ ¼ 3:58N 2 18:16N þ 59
ð34Þ
7
It can then be deduced from (34) that a reasonably conditioned matrix with C < 10 is obtained for a spatial
resolution as high as Oð1000Þ. The method can therefore be applied to problems where small-scale field features require use of high spatial resolution.
6. Generalizations
We now discuss the applicability of this method to other geometries, problems, and spatial discretizations. The decomposition into interior particular and homogeneous functions and exterior homogeneous
functions described in Section 2 is, of course, completely general and not related to any particular spatial
discretization. The method described in Section 3.1 for constructing the exterior harmonic functions relies
on the Chebyshev–Chebyshev discretization of the rectangle which is widely used since the Chebyshev polynomials are optimal approximants of smooth functions. This property, as well as the straightforward correspondence between interior and exterior solutions, make the Chebyshev discretization especially suitable
for the construction of the exterior harmonic solutions as well. However, the method is easily generalizable
to other basis functions fk(x) for the potential values, which can be substituted into (25) in order to calculate
the corresponding charge densities rk(x), if dictated by the geometry or numerical method used for the interior problem.
In fact, since our real interest is in generating the complete set of rk(x) necessary to generate the complete
set of /k(x), rather than in calculating the specific rk(x)pffiffiffiffiffiffiffiffiffiffiffiffi
corresponding
to each particular fk(x), the only
ffi
information really required in (26) is the singularity 1=p 1 x2 . One may then allow the set of rk’s to be
the products of this singularity with the members of any appropriate basis set of analytic functions on the
boundary in question.
In three dimensions, the fundamental Green’s function is
Gðx; x0 Þ ¼
1
1
4p jx x0 j
ð35Þ
In an axisymmetric geometry with a Fourier representation of the azimuthal direction, all of the problems to
be solved decouple according to Fourier mode. The operation count would then scale linearly with the number
of azimuthal points or Fourier modes. The elliptic problems in (6) would remain two-dimensional, and the
integral equations equivalent to (24) would remain one-dimensional.
This method can also be applied to other elliptic problems or to parabolic problems. As stated in Section 1,
our motivation for developing this method is to apply it to the magnetohydrodynamic Eqs. (1)–(3), in which
(1) is a parabolic equation. A general parabolic problem may be written as:
ot U ¼ DU þ FðU; qÞ
ð36Þ
where F may include nonlinear and/or time-dependent source terms. First-order implicit temporal discretization of (36) results in the inhomogeneous Helmholtz equation:
ðI dtDÞUðt þ dtÞ ¼ F
ð37Þ
where F may depend on previous values of U. This Helmholtz operator ðI dtDÞ can replace the Laplacian in
(4) and (6). It is known that replacement of the Helmholtz equation
by a boundary integral equation can lead
pffiffiffiffi
to singularities for certain values of the wavenumber (here i= dt); a large body of work, e.g. [37–41], addresses
this problem. However, in the magnetohydrodynamic case of a conducting fluid surrounded by an exterior
vacuum, no such difficulties would be introduced, since the exterior problem remains governed by Laplace’s
equation. More complicated vectorial operators may appear, as occur in the Navier–Stokes or magnetohydrodynamic equations.
7. Comparison with other approaches
We mention here some other techniques that have been used to solve exterior problems or to match interior
and exterior domains. Spectral methods can be combined with various transformations and mappings. The
P. Boronski / Journal of Computational Physics 225 (2007) 449–463
461
inner region can be surrounded by a sphere, and the outer domain decomposed into the region inside and outside the sphere. The exterior domain can be mapped into an interior domain via a 1/r mapping [6,42,43]; spectral methods can then be used to treat either or both domains. The region exterior to one or more spheres has
been mapped to the interior of a rectangle [44] or a pentangular [45] region rotated about an axis, and Chebyshev–Fourier expansions used to solve elliptic equations arising in the study of black holes in general relativity.
A smooth boundary can be parameterized by angle, and the boundary values represented as a series of trigonometric functions or spherical harmonics [22,39–43,46–48]. Our method differs from these in that a Chebyshev approximation is used to represent the boundary values on each segment of a non-smooth boundary, and
an analytic formula is used to calculate the surface density which exactly yields this Chebyshev approximation.
Conformal mapping is another technique which can be used to calculate interior or exterior harmonic functions. The Riemann mapping theorem guarantees the existence of a conformal transformation from the interior or exterior of a simply connected domain into the interior or exterior of a unit disk; its proof is, however,
non-constructive, and does not explicitly derive the transformation. For some geometries, including the exterior of a rectangle, an analytical formula can be derived [49]. For polygon-bounded regions with piecewiseconstant boundary conditions, the Schwarz–Christoffel [50] mapping has proved to be a very robust tool,
applied to problems arising in magneto- and electro- statics, potential flows, inverse problems and many other
fields.
Our influence matrix approach relies on calculating harmonic functions with arbitrary boundary data, for
which conformal mapping is much more problematic. More general conformal mappings are often computed
by solving Symm’s or Carleman’s equation (24) numerically on the domain boundary, making this approach
similar in terms of numerical cost and precision to boundary integral equation methods. It is interesting to
note that, for domains including corners, the Chebyshev approximation is especially well suited, guaranteeing
superconvergence of the mapping function [33].
8. Conclusion
As a test case for the magnetohydrodynamic equations, we have developed a method for solving the twodimensional Poisson equation in a bounded domain, where the solution satisfies matching conditions with a
harmonic potential outside the domain. The method solves only the interior problem and determines the
boundary conditions ensuring smooth matching with the exterior solution. The essential element of this
approach is construction of a basis of harmonic functions which represent the near-boundary exterior solutions uniformly. This basis is used to construct the influence matrix which serves to impose the coupled boundary conditions between the interior and exterior solutions. The method is numerically reasonably well
conditioned and can be used for quite high spatial resolutions. For a spectral solver, this method guarantees
exponential convergence.
Instead of corresponding to point sources on the boundary, each exterior harmonic solution corresponds to
a spectral basis function. The most costly process – the construction of a basis of exterior harmonic functions –
depends only on geometry and spatial resolution. Once the basis is computed it can be stored and used for any
computation using the same resolution and domain shape. When used as a preprocessing step for time-dependent simulations, the cost of constructing the exterior harmonic basis is negligible compared to that of thousands of time steps.
Since cylindrical coordinates have one periodic direction, it should be possible to apply this method separately to each of the Fourier modes, treated individually as two-dimensional problems. The extension of this
method to the MHD equations in a finite cylindrical geometry is currently under investigation.
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